This page presents 4 programs related to sample size requirements when comparing 2 means, and tables of these sample sizes.

The programs and tables on this page assumes that the measurements are continuous and normally distributed.

The following 4 programs are available on this page

• Sample Size requires the following input
  • The probability of Type I Error (p, &alphs;) that will be used to determine significance. This is usually 0.05, but 0.1, 0.01, and 0.001 are also used
  • The power (1 - β) required. Usually this is 0.8, and sometimes 0.9
  • The critical difference, the nominated or expected difference in means the model is to detect
  • The nominated or anticipated within group Standard Deviation of the population being studied
  The results is the sample size in each of the 2 groups, assuming that they are equal. Sample size for the 1 and 2 tail models are produced.

  For example, if α of 0.05 and power of 0.8 are required, and the difference to detect is half (0.5) of the expected Standard Deviation, then the sample size required is 51 cases per group for one tail comparison and 64 for two tail.

• Power requires the following imput
  • The probability of Type I Error (p, &alphs;) that will be used to determine significance. This is usually 0.05, but 0.1, 0.01, 1nd 0.001 are also used
  • The difference between the two means observed
  • The sample size and Standard Deviation observed in the 2 groups (n1, SD1) and (n2, SD2)
  The results are the powers of the comparison, 1 and 2 tail models.

  For example, in a comparison of birth weight between boys and girls, using α of 0.05 is to be used to determine statistical significance, the difference betwwen the means is 200 grams, 50 boys with Standard Deviations of 400 grams and 60 girls with Standard Deviation of 380 grams. The power is 0.85 for a 1 tail test and 0.77 two tail.

• Confidence Interval (CI) reuires the following input
  • The percentage representing the level of confidence reqwuired. This is usually 95%, but sometimes 99% is used
  • The sample size and Standard Deviation found in group 1 (n1, SD1) and group 2 (n2, SD2)
  The results are the confidence interval of the difference, 1 and 2 tail

  For example, if we use the same results as that in the power program, and wish to know the 95% confidence intervals of the difference. n1 = 50, SD1 = 400, n2 = 60, SD2 = 380. The results are ±123.6 grams for 1 tail and ±147.7 grams for 2 tail

  With the difference found to be 200 grams

  • The 1 tail confidence interval is >176.4 or <323.6. So if the question is whether girls are lighter than boys, we can conclude that boys are at least 176.4 grams heavier than girls
  • The 2 tail confidence interval is 162.3 to 347.7 grams. So if the question is whther boys and girls differ in birth weight, without asking which is greater, the answer is no less than 162.3 grams and no greater than 347.7 grams
• Pilot Studies requires the following parameters

  	1 Tail				2 Tail
  Ssiz	CI1	Diff	Dec/case	%Dec/cas	CI	Diff	Dec/case	%Dec/cas
  5	2.3522				2.9169
  10	1.551	0.8012	0.1602	6.8	1.8791	1.0378	0.2076	7.1
  15	1.2423	0.3087	0.0617	4	1.496	0.3832	0.0766	4.1
  20	1.0663	0.1761	0.0352	2.8	1.2803	0.2156	0.0431	2.9
  25	0.9488	0.1175	0.0235	2.2	1.1374	0.1429	0.0286	2.2
  30	0.8632	0.0856	0.0171	1.8	1.0337	0.1037	0.0207	1.8
  35	0.7973	0.0659	0.0132	1.5	0.954	0.0797	0.0159	1.5
  40	0.7444	0.0528	0.0106	1.3	0.8903	0.0637	0.0127	1.3
  45	0.7009	0.0435	0.0087	1.2	0.8379	0.0524	0.0105	1.2
  50	0.6642	0.0367	0.0073	1	0.7938	0.0441	0.0088	1.1
  55	0.6327	0.0315	0.0063	0.9	0.756	0.0378	0.0076	1
  60	0.6054	0.0274	0.0055	0.9	0.7231	0.0329	0.0066	0.9
  65	0.5813	0.0241	0.0048	0.8	0.6942	0.0289	0.0058	0.8
  70	0.5598	0.0214	0.0043	0.7	0.6685	0.0257	0.0051	0.7
  75	0.5406	0.0192	0.0038	0.7	0.6454	0.0231	0.0046	0.7
  80	0.5232	0.0174	0.0035	0.6	0.6246	0.0208	0.0042	0.6
  85	0.5074	0.0158	0.0032	0.6	0.6057	0.0189	0.0038	0.6
  90	0.493	0.0144	0.0029	0.6	0.5883	0.0173	0.0035	0.6
  95	0.4797	0.0133	0.0027	0.5	0.5725	0.0159	0.0032	0.5
  100	0.4674	0.0123	0.0025	0.5	0.5578	0.0147	0.0029	0.5
  105	0.4561	0.0114	0.0023	0.5	0.5442	0.0136	0.0027	0.5

  • The probability of Type I Error (p, &alphs;) that will be used to determine significance. This is usually 0.05 or 0.1
  • The expected within group, population Standard Deviation of the study
  • The interval of sample size per group (intv) to exaamine changes in confidence interval as sample sizw=es increase. Usually this is between 3 and 10 cases
  • The maximum sample size per group for the estimates. In most cases, pilot studies end in 30 to 40 cases, and there is no point having a pilot study with more than 100 cases per group. A common value used is 50
  The program produces a table as shown to the right, nominationg a Standard Deviation of 1, and 95% confidence interval for the pilot. With increasing sample size, the reduction in confidence interval decreases, and it can be seen that beyond 30 cases per group, the further decrease in confidence interval become trivial. A conclusion can therefore be made that a sample size of 30 cases per group would be suitable for a pilot study, to define the expected Standard Deviation and difference in the 2 means, as well as providing information on the research environment, so that a formal comparison can be planned.

References

Armitage P (1980) Statistics in Medical Research. Blackwell Scientific Publication, Oxford. ISBN 0 632 05430 1 p.120

Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 24-25

Johanson GA and Brooks GP (2010) Initial Scale Development: Sample Size for Pilot Studies. Educational and Psychological Measurement Vol.70,Iss.3;p.394-400

These tables provides sample sizes (per group) needed to compare the means of 2 groups, assuming that the data is normally distributed, that the variance in the groups are similar, and that the sample sizes for the groups are the same
   - Power = 1-β
   - α = Probability of Type I Error
   - es = Effect size = difference between means / population of within group Standard Deviation
   - column header 1T and 2T represents 1 and 2 tail
   - sample size is that for each of the 2 groups
Effect Size 0.05 to 1 Effect Size 1.05 to 2 Effect Size 2.05 to 3

Data

Sample Size: Data a table of 4 columns   - Each row contains data from a separate study   - Col 1 = Probability of Type I Error α   - Col 2 = Power (1 - β)   - Col 3 = Difference Between the Two Means to be Detected   - Col 4 = Expected Within Group Standard Deviation       Power: Data a table of 6 columns   - Each row contains data from a separate study   - Col 1 = probability of Type I error (α)   - Col 2 = Differences between group means observed   - Col 3, 5 = sample size used in Group 1 & 2   - Col 4, 6 = Observed Standard Deviation of Group 1 & 2       Confidence Interval: Data a table of 5 columns   - Each row contains data from a separate study   - Col 1 = Percent confidence (usually 95 or 99)   - Col 2, 4 = sample size group 1 & 2   - Col 3, 5 = Standard Deviation group 1 & 2       Pilot Study: Data is for a single plan, a single column with 4 rows   - Row 1 = Percent confidence required, usually 95 or 99   - Row 2 = within Group or pupulation Standard Deviation   - Row 3 = Sample size intervals   - Row 4 = Maximum sample size

This subpanel presents 4 R programs related to sample size for 2 unpaired means

Program 1: Estimating sample size

# Pgm 1: Sample Size
# data entry
dat = ("
Alpha   Power     Diff    SD
0.05	0.8	  0.5	  1.0
0.01	0.8	  0.5	  1.0
0.05	0.9	  0.5	  1.0
0.01	0.9	  0.5	  1.0
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    

# vectors to store results
SSiz1Tail <- vector()
SSiz2Tail <- vector()

# Calculations
delta <- abs(df$Diff / df$SD)            # effect size
zb <- abs(qnorm(1 - df$Power))           # z for beta
# 1 tail
za <- abs(qnorm(df$Alpha))               # 1 tail z for alpha 
f <-  (za + zb) / delta
SSiz1Tail <- append(SSiz1Tail,ceiling(2.0 * f**2 + za**2 / 4.0))  # 1 tail sample size
# 2 tail
za <- abs(qnorm(df$Alpha / 2))           # 2 tail z for alpha 
#za
f <-  (za + zb) / delta
SSiz2Tail <- append(SSiz2Tail,ceiling(2.0 * f**2 + za**2 / 4.0))  # 2 tail sample size
# append results to data frame
df$SSiz1Tail <- SSiz1Tail
df$SSiz2Tail <- SSiz2Tail
df # show data frame with input data and rsults

The results are as follows. Sample size is for each group

> df # show data frame with input data and rsults
  Alpha Power Diff SD SSiz1Tail SSiz2Tail
1  0.05   0.8  0.5  1        51        64
2  0.01   0.8  0.5  1        82        96
3  0.05   0.9  0.5  1        70        86
4  0.01   0.9  0.5  1       106       121

Program 2: Power

# Pgm2: Power
# data entry
dat = ("
Alpha Diff  N1 SD1  N2   SD2
0.05	0.5	64	1.0	64	1.0
0.01	0.5	96	1.0	96	1.0
0.01	0.5	64	1.0	96	1.0
0.05	0.5	86	1.0	86	1.0
0.01	0.5	121	1.0	121	1.0
0.01	0.5	86	1.0	121	1.0
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame  
 
# vector to store results
Power1Tail <- vector()
Power2Tail <- vector()

# Calculations
se <- sqrt(((df$N1 - 1) * df$SD1**2 + (df$N2 - 1) * df$SD2**2) / (df$N1 + df$N2 - 2))
n <- (df$N1 + df$N2) / 2 
delta <- abs(df$Diff / se)
# 1 tail
za <- abs(qnorm(df$Alpha))                  # 1 tail z for alpha 
zb <- delta * sqrt(n / 2.0 - za / 8.0) - za
Power1Tail <- append(Power1Tail,pnorm(zb))         # power 1 tail
# 2 tail
za <- abs(qnorm(df$Alpha / 2))               # 2 tail z for alpha 
zb <- delta * sqrt(n / 2.0 - za / 8.0) - za
Power2Tail <- append(Power2Tail,pnorm(zb))         # power 2 tail

# append ewsults to data frame
df$Power1Tail <- Power1Tail
df$Power2Tail <- Power2Tail
df # show data frame with input data and results

The results are as follows

> df # show data frame with input data and results
  Alpha Diff  N1 SD1  N2 SD2 Power1Tail Power2Tail
1  0.05  0.5  64   1  64   1  0.8798970  0.8044474
2  0.01  0.5  96   1  96   1  0.8701805  0.8096574
3  0.01  0.5  64   1  96   1  0.7951479  0.7169130
4  0.05  0.5  86   1  86   1  0.9480270  0.9048008
5  0.01  0.5 121   1 121   1  0.9398340  0.9036948
6  0.01  0.5  86   1 121   1  0.8962385  0.8437134

Program 3: Confidence Interval

# Pgm3: Confidence Interval

#                 subroutine to calculate confidence interval
# subroutine to calculate confidence interval
CalCI <- function(alpha, n1, sd1, n2, sd2)
{
  se = sqrt(((n1 - 1) * sd1**2 + (n2 - 1) * sd2**2) / 
               (n1 + n2 - 2) * (1 / n1 + 1 / n2))
  degfm = n1 + n2 - 2
  t1 = qt(1 - alpha, degfm)
  ci1 = t1*se
  t2 = qt(1 - alpha / 2, degfm)
  ci2 = t2 * se
  return (c(se, ci1, ci2))
}

The main program for confidence interval, using the subroutine CalCI

# data entry
dat = ("
       Pc    N1   SD1   N2   SD2
95	100	18.5	100	16.8
99	100	18.5	100	16.8
95	50	8.5	30	5.2
99	50	8.5	30	5.2
95	50	400	60	380
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame  
# df
# vectors for results
SE <- vector()   # Standard Error
CI1 <- vector()  # Confidence interval 1 tail
CI2 <- vector()  # Confidence interval 2 tail
# calculations
# 1 tail
for(i in 1 : nrow(df))
{
  alpha <- (1 - df$Pc[i] / 100)     # alpha 1 tail
  ar <- CalCI(alpha,df$N1[i], df$SD1[i], df$N2[i], df$SD2[i])
  #ar <- CalCI(1 - df$Pc[i] / 100,df$N1[i], df$SD1[i], df$N2[i], df$SD2[i])
  SE <- append(SE, ar[1])    # Standard Error
  CI1 <- append(CI1, ar[2])  # Confidence interval 1 tail
  CI2 <- append(CI2, ar[3])  # Confidence interval 2 tail
}
df$SE <- SE
df$CI1 <- CI1
df$CI2 <- CI2
df

The results are as follows

• CI1 and CI2 are confidence intervals for 1 and 2 tails
• The interval is distance from mean to the limit, and should be represented as mean ± CI

> df # Data frame withinput data and results

  Pc  N1   SD1  N2   SD2        SE        CI1        CI2
1 95 100  18.5 100  16.8  2.498980   4.129778   4.928032
2 99 100  18.5 100  16.8  2.498980   5.860928   6.499565
3 95  50   8.5  30   5.2  1.719553   2.862410   3.423366
4 99  50   8.5  30   5.2  1.719553   4.084128   4.540204
5 95  50 400.0  60 380.0 74.526406 123.645653 147.724266

Program 4: Pilot study

Please Note: This program shares the same subroutine as Program 3 to calculate SE and CI, but the subroutine is not duplicated here

# Program 4. Pilot study
# Parameters
pc = 95         # % confidence
sd = 1.0        # within group or population SD
intv = 5        # interval
maxN = 100      # maximum sample size
# vectors for results
SSiz <- vector()     # sample size
CI1 <- vector()      # confidence interval 1 tail
Diff1 <- vector()    # difference in CI from previous row 1 tail
DecCase1 <- vector() # decrease in CI per case increase 1 tail
PDCase1 <- vector()  # % decrease in CI per case increase 1 tailCI1 <- vector()      # confidence interval 1 tail
CI2 <- vector()      # confidence interval 2 tail
Diff2 <- vector()    # difference in CI from previous row 2 tail
DecCase2 <- vector() # decrease in CI per case increase 2 tail
PDCase2 <- vector()  # % decrease in CI per case increase 2 tail
# Calculations

alpha = 1 - pc / 100
# first row
n = intv
SSiz <- append(SSiz,n)
ar <- CalCI(alpha, n, sd, n, sd)
ci1 = ar[2] * 2
CI1 <- append(CI1,sprintf(ci1, fmt="%#.4f"))      # confidence interval 1 tail
Diff1 <- append(Diff1,0)    # difference in CI from previous row 1 tail
DecCase1 <- append(DecCase1,0) # decrease in CI per case increase 1 tail
PDCase1 <- append(PDCase1,0)  # % decrease in CI per case increase 1 tailCI1 <- vector()      # confidence interval 1 tail
ci2 = ar[3] * 2
CI2 <- append(CI2,sprintf(ci2, fmt="%#.4f"))      # confidence interval 1 tail
Diff2 <- append(Diff2,0)    # difference in CI from previous row 1 tail
DecCase2 <- append(DecCase2,0) # decrease in CI per case increase 1 tail
PDCase2 <- append(PDCase2,0)  # % decrease in CI per case increase 1 tailCI1 <- vector()      # confidence interval 1 tail
# subsequent rows
while(n < maxN)
{
  n = n + intv
  SSiz <- append(SSiz,n)
  ar <- CalCI(alpha, n, sd, n, sd)
  oldci1 = ci1
  ci1 = ar[2] * 2
  CI1 <- append(CI1,sprintf(ci1, fmt="%#.4f"))      # confidence interval 1 tail
  diff1 = oldci1 - ci1
  Diff1 <- append(Diff1,sprintf(diff1, fmt="%#.4f"))    # difference in CI from previous row 1 tail
  decCase1 = diff1 / intv
  DecCase1 <- append(DecCase1,sprintf(decCase1, fmt="%#.4f")) # decrease in CI per case increase 1 tail
  pDCase1 = sprintf(decCase1 / oldci1 * 100, fmt="%#.1f")
  PDCase1 <- append(PDCase1,pDCase1)  # % decrease in CI per case increase 1 tail

  oldci2 = ci2
  ci2 = ar[3] * 2
  CI2 <- append(CI2,sprintf(ci2, fmt="%#.4f"))      # confidence interval 2 tail
  diff2 = oldci2 - ci2
  Diff2 <- append(Diff2,sprintf(diff2, fmt="%#.4f"))    # difference in CI from previous row 2 tail
  decCase2 = diff2 / intv
  DecCase2 <- append(DecCase2,sprintf(decCase2, fmt="%#.4f")) # decrease in CI per case increase 2 tail
  pDCase2 = sprintf(decCase2 / oldci2 * 100, fmt="%#.1f")
  PDCase2 <- append(PDCase2,pDCase2)  # % decrease in CI per case increase 2 tail
}
df <- data.frame(SSiz,CI1,Diff1,DecCase1,PDCase1,CI2,Diff2,DecCase2,PDCase2)
df # display results in data frame

The results are as follows. Please note:

• The width of sample size is twicw that in program 3 because it is the sum from both sides of the mean
• SSiz = sample size
• CI1 and CI2 = confidence intervals (1 and 2 tail)
• Diff1 and Diff 2 = difference in confidence interval from the previous row (1 and 2 tail)
• CaseDec1 and CaseDec2 = change in confidence interval per case increase (1 and 2 tail)
• PcDec1 and PcDec2 = % change in confidence interval per case increase (1 and 2 tail)

> df # display results in data frame
   SSiz    CI1  Diff1 DecCase1 PDCase1    CI2  Diff2 DecCase2 PDCase2
1     5 2.3522      0        0       0 2.9169      0        0       0
2    10 1.5510 0.8012   0.1602     6.8 1.8791 1.0378   0.2076     7.1
3    15 1.2423 0.3087   0.0617     4.0 1.4959 0.3832   0.0766     4.1
4    20 1.0663 0.1760   0.0352     2.8 1.2803 0.2156   0.0431     2.9
5    25 0.9488 0.1175   0.0235     2.2 1.1374 0.1430   0.0286     2.2
6    30 0.8632 0.0856   0.0171     1.8 1.0337 0.1037   0.0207     1.8
7    35 0.7973 0.0659   0.0132     1.5 0.9540 0.0797   0.0159     1.5
8    40 0.7444 0.0528   0.0106     1.3 0.8903 0.0637   0.0127     1.3
9    45 0.7009 0.0435   0.0087     1.2 0.8379 0.0524   0.0105     1.2
10   50 0.6642 0.0367   0.0073     1.0 0.7938 0.0441   0.0088     1.1
11   55 0.6328 0.0315   0.0063     0.9 0.7560 0.0378   0.0076     1.0
12   60 0.6054 0.0274   0.0055     0.9 0.7231 0.0329   0.0066     0.9
13   65 0.5813 0.0241   0.0048     0.8 0.6942 0.0289   0.0058     0.8
14   70 0.5598 0.0214   0.0043     0.7 0.6685 0.0257   0.0051     0.7
15   75 0.5406 0.0192   0.0038     0.7 0.6454 0.0231   0.0046     0.7
16   80 0.5232 0.0174   0.0035     0.6 0.6246 0.0208   0.0042     0.6
17   85 0.5074 0.0158   0.0032     0.6 0.6057 0.0189   0.0038     0.6
18   90 0.4930 0.0144   0.0029     0.6 0.5883 0.0173   0.0035     0.6
19   95 0.4797 0.0133   0.0027     0.5 0.5724 0.0159   0.0032     0.5
20  100 0.4674 0.0123   0.0025     0.5 0.5578 0.0147   0.0029     0.5